Abstract:
The spectral element method is used to discretize self-adjoint elliptic
equations in three dimensional domains. The domain is decomposed into
hexahedral elements, and in each of the elements the discretization
space is the set of polynomials of degree $N$ in each variable. A
conforming Galerkin formulation is used, the corresponding integrals are
computed approximately with Gauss-Lobatto-Legendre (GLL) quadrature rules
of order $N$, and a Lagrange interpolation basis associated with the GLL
nodes is used. Fast methods are developed for solving the resulting
linear system by the preconditioned conjugate gradient method.
The conforming {\it finite element} space on the GLL mesh, consisting
of piecewise $Q_{1}$ or $P_1$ functions, produces a stiffness matrix
$K_h$ that is known to be spectrally equivalent to the spectral element
stiffness matrix $K_N$. $K_h$ is replaced by a preconditioner
$\tilde{K}_h$ which is well adapted to parallel computer architectures.
The preconditioned operator is then $\tilde{K}_h^{-1} K_N$.

Our techniques for non-regular meshes make it possible to estimate the
condition number of $\tilde{K}_h^{-1} K_N$, where $\tilde{K}_h$ is a
standard finite element preconditioner of $K_h$, based on the GLL mesh.
The analysis of two finite element based preconditioners: the wirebasket
method of Smith, and the overlapping Schwarz algorithm for the spectral
element method, are given as examples of the use of these tools.
Numerical experiments performed by Pahl are briefly discussed
to illustrate the efficiency of these methods in two dimensions.